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After studding this lecture, the student should be able

to: define the reason for slowing down neutrons;

explain what nuclear reaction is the best for slowing down;

explain main principals of elastic scattering —

post collision energy range, frequency function,

average energy loss per one collision;

define lethargy of neutrons and its connection to energy.

First I should explain the typical neutron cycle

in a nuclear reactor.

By default for simplification reasons

we will consider the thermal reactor.

Let us imagine that in a core in the moment N1

neutrons were born momentarily by a fission, thus they are fast.

What can happen with the fast neutrons —

first, a part of them can fly out from the reactor core,

second the fission reaction can occur

for any heavy nuclei — U235 or 238.

The next is the process of neutron slowing down

and during the process

the neutrons can suffer the resonance absorption.

When the neutrons become thermal

they finish their life by absorbing

in for example non-fuel absorbent – it is bad,

or in the fuel but non-fissile reaction occurs — it is bad too,

and finally they can produce the fissions —

we get 200 MeV energy release

and 2 or 3 additional neutron for chain reaction.

Let the number of neutrons after the whole cycle is N2.

N1 and N2 are the initial and subsequent generations of neutrons.

Thus, k = N2/N1 is the multiplication factor

of the system (reactor),

which takes into consideration all neutron processes.

For infinite homogeneous medium

this definition of the multiplication factor is the same up

to factor of nu_f Sigma_f

and the absorption macroscopic cross section.

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Please note, that to slow down or not it is our decision.

It is our choice of reactor core design.

The reason for slowing down lies in the dependency

of microscopic cross section of fission on energy.

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You can see, that in the large energy range

sigma_f equals approximately several barns,

but in the thermal range sigma_f equals hundreds barns.

That means in the thermal range the probability

to make fission is much higher

compared to large energy range.

In turn, we need less uranium fuel for thermal reactors,

they are more simple and cheaper as compared to fast reactors.

In this part we should consider main laws

of the neutron slowing down processs

and, second, investigate dependency

of the neutron flux on the energy —

the latter one is called the neutron spectrum.

How can we slow down the neutrons?

Of course, by nuclear reactions of interaction

of neutrons with the nuclei of a medium.

We know two types of the nuclear reactions —

absorption and scattering.

All these types have an influence on the neutron spectrum.

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The absorption reaction influence means that

after the fission the fast neutrons have

certain energy distribution called the fission spectrum,

the average neutron energy is about 2 MeV.

The main process of slowing down is the scattering.

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The resonance inelastic scattering

can moderate the neutrons.

It is important, but it is not the main process.

Because resonance levels for light nuclei

are in the high energy range

and for heavy nuclei the lowest energy level

is in the range of keV. It's not enough.

The resonance elastic scattering doesn’t moderate neutrons.

So, the main nuclear reaction to slow down neutrons

is the potential elastic scattering.

Look at the typical microscopic cross section

of the elastic scattering.

As usual there are three regions in the dependency

microscopic cross section of scattering

on energy in epithermal energy range.

The first is a tableland region —

this is the potential elastic scattering.

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It is limited from the right

by the resonance zone

excluding hydrogen and deuterium.

For light nuclei the potential elastic scattering

takes place till hundreds of keV.

In contrast to heavy nuclei, for example U-238,

the right border is in the range of eV.

We can define the slowing down energy range —

the following set of laws,

formulas and equations will be true

for the region and it will be our main approximation.

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The slowing down range is limited from the left

by the thermal range (about 1 eV)

and from the right, first,

by the border of potential scattering

and, second, by the isotropy condition

of the potential elastic scattering

in the center of mass system.

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There is an assessment formula to define the energy

of beginning of the potential scattering anisotropy —

E max equal to 10 MeV divided by the mass of nucleus

designeted A in the power 3/2.

So, for example, for the hydrogen

there is no anisotropy energy range.

Thus, eventually, the right border of slowing down range

is about 100 keV.

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There are two additional assumptions to be continued.

First, neutron impact to single nucleus

neutron doesn’t feel the nuclei binding

in a molecule or a crystal lattice.

Second, before scattering

the nucleus is at rest in the lab system of coordinates.

That means the neutron speed is much bigger compared

to the thermal oscillatory movement of a nucleus.

This assumption is true for the neutron energy above 20 meV.